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The question presented involves finding probabilities using the normal distribution. Let's break it down step by step for both parts (a) and (b).

### Given Information:
- Mean (\(\mu\)) = 33.2
- Standard deviation (\(\sigma\)) = 68.9
- Sample size for part (b) (\(n\)) = 14

### Part (a) - Find \( P(X > 57.1) \):
Here, we are asked to find the probability that a single randomly selected value from a normal distribution exceeds 57.1.

1. We need to calculate the z-score using the formula:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   Where:
   - \( X = 57.1 \)
   - \( \mu = 33.2 \)
   - \( \sigma = 68.9 \)

2. After finding the z-score, we can look up the corresponding probability from the standard normal distribution table or use a calculator.

### Part (b) - Find \( P(M > 57.1) \):
This part asks for the probability that the mean of a sample of size \( n = 14 \) is greater than 57.1.

1. We use the **sampling distribution** of the sample mean, which has the same mean (\(\mu = 33.2\)) but a smaller standard deviation (called the **standard error**) given by:
   \[
   \text{Standard Error} = \frac{\sigma}{\sqrt{n}}
   \]
   So:
   \[
   \text{Standard Error} = \frac{68.9}{\sqrt{14}}
   \]

2. Now, calculate the z-score for the sample mean:
   \[
   Z = \frac{M - \mu}{\text{Standard Error}}
   \]
   Where:
   - \( M = 57.1 \)

3. After finding this z-score, we again use the standard normal distribution table or calculator to find the probability.

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Let me calculate both parts now.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you on how to calculate this by hand or through another tool. Let me know if you would like me to walk you through the steps manually!

Here are a few related questions to think about:

1. How do you calculate the z-score for any given value in a normal distribution?
2. What is the significance of the standard error in sampling distributions?
3. How do you find probabilities using the standard normal distribution table?
4. Why do larger sample sizes reduce the standard error?
5. What changes in probability when you switch from a single value to a sample mean?

**Tip**: For normal distribution problems, always check if the question deals with individual values or sample means, as this will affect your standard deviation.

A population of values has a normal distribution with µ = 33.2 and σ = 68.9. If a random sample of size n = 14 is selected, find: (a) The probability that a single randomly selected value is greater than 57.1. (b) The probability that a sample of size n = 14 is randomly selected with a mean greater than 57.1.

This problem involves calculating probabilities using a normal distribution with a given population mean (µ = 33.2) and standard deviation (σ = 68.9). It covers two key tasks: finding the probability that a single value exceeds 57.1, and calculating the probability for a sample mean in a sample size of 14. Suitable for students learning about Z-scores and standard errors.

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